Theorem dfdm3 4909

Description: Alternate definition of domain. Definition 6.5(1) of [TakeutiZaring] p. 24. (Contributed by NM, 28-Dec-1996.)

Assertion

Ref	Expression
dfdm3	|- dom A = { x | E. y <. x , y >. e. A }

Distinct variable group:   x, y, A

Proof of Theorem dfdm3

Step	Hyp	Ref	Expression
1		df-dm 4729	. 2 |- dom A = { x | E. y x A y }
2		df-br 4084	. . . 4 |- ( x A y <-> <. x , y >. e. A )
3	2	exbii 1651	. . 3 |- ( E. y x A y <-> E. y <. x , y >. e. A )
4	3	abbii 2345	. 2 |- { x | E. y x A y } = { x | E. y <. x , y >. e. A }
5	1, 4	eqtri 2250	1 |- dom A = { x | E. y <. x , y >. e. A }

Syntax hints:   = wceq 1395   E.wex 1538   e. wcel 2200   {cab 2215   <.cop 3669   class class class wbr 4083   dom cdm 4719

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-11 1552  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-br 4084  df-dm 4729

This theorem is referenced by:  csbdmg  4917